Question

semester exam review-algebra
graph the function and its parent function.
then describe the transformation.

1. $g(x) = 3|x + 2| + 3$

Explanation:

Step1: Identify Parent Function

The parent function of absolute - value functions is \(y = |x|\). Its graph is a V - shaped graph with the vertex at the origin \((0,0)\), opening upwards, and passing through points like \((1,1)\) and \((- 1,1)\).

Step2: Analyze Transformations for \(g(x)=3|x + 2|+3\)

• Horizontal Shift: For the absolute - value function \(y = |x - h|\), the graph shifts \(h\) units horizontally. In \(g(x)=3|x+2| + 3=3|x-(-2)|+3\), the value of \(h=-2\). So, the graph of \(y = |x|\) is shifted 2 units to the left.
• Vertical Stretch: The coefficient \(a\) in \(y=a|x - h|+k\) affects the vertical stretch or compression. Here, \(a = 3>1\), so the graph of \(y=|x + 2|\) (after the horizontal shift) is vertically stretched by a factor of 3.
• Vertical Shift: The value of \(k\) in \(y=a|x - h|+k\) affects the vertical shift. Here, \(k = 3\), so the graph of \(y = 3|x + 2|\) (after the horizontal shift and vertical stretch) is shifted 3 units up.

Step3: Graphing the Parent Function and \(g(x)\)

• Graph of Parent Function (\(y = |x|\)):
• Plot the vertex at \((0,0)\).
• For \(x>0\), the slope is 1. So, when \(x = 1\), \(y = 1\); when \(x=2\), \(y = 2\), etc.
• For \(x<0\), the slope is - 1. So, when \(x=-1\), \(y = 1\); when \(x = - 2\), \(y=2\), etc. Draw a V - shaped graph.
• Graph of \(g(x)=3|x + 2|+3\):
• Vertex: From the transformations, the vertex of \(g(x)\) is at \((-2,3)\) (because of the horizontal shift left by 2 units and vertical shift up by 3 units).
• Vertical Stretch: Since the vertical stretch factor is 3, for a given \(x\) - value relative to the vertex, the \(y\) - value is 3 times the \(y\) - value of \(y = |x+2|\) at that \(x\) - value. For example, when \(x=-2\), \(g(-2)=3| - 2 + 2|+3=3\). When \(x=-1\), \(g(-1)=3|-1 + 2|+3=3\times1 + 3=6\). When \(x=-3\), \(g(-3)=3|-3 + 2|+3=3\times1+3 = 6\). When \(x = 0\), \(g(0)=3|0 + 2|+3=3\times2+3=9\). When \(x=-4\), \(g(-4)=3|-4 + 2|+3=3\times2+3 = 9\). Plot these points and draw a V - shaped graph with the vertex at \((-2,3)\), opening upwards, with a steeper slope (due to the vertical stretch) compared to the parent function.

Answer:

The parent function is \(y = |x|\). The function \(g(x)=3|x + 2|+3\) is obtained by shifting the graph of \(y = |x|\) 2 units to the left, vertically stretching it by a factor of 3, and then shifting it 3 units up. The graph of \(y = |x|\) has a vertex at \((0,0)\) and a V - shape with slope 1 for \(x>0\) and - 1 for \(x<0\). The graph of \(g(x)\) has a vertex at \((-2,3)\), is vertically stretched, and shifted left and up from the parent function. The transformation of \(g(x)\) from \(y = |x|\) is a horizontal shift 2 units left, vertical stretch by factor 3, and vertical shift 3 units up.